Trading Options

• Q. What do you trade?

  A. I trade stocks and stock options. This is where much of research is focused and resources are devoted. Foreign currency futures are traded to manage the exposure to the value of the US dollar. Commodity futures have also been traded infrequently during the life of the portfolio.

• Q. What was your return in 2006?

  A. 28.6% for the full portfolio. The stock and option portfolio alone earned 22.9%; this should be viewed as a part of the return of the full portfolio. The difference is due to trades in futures.

• Q. What was the total return of the S&P 500 index in 2006?

  A. 15.6%, including 1.79% estimated contribution from dividend payments of the constituent stocks.

• Q. What was the standard deviation of your daily returns in 2006?

  A. 1.29% for the full portfolio. If only trades in stocks and options are considered, the figure is only 0.79%. The difference is due to contribution to daily P&L of positions in futures.

• Q. What was the standard deviation of daily returns of the S&P 500 index in 2006?

  A. 0.63%

Why such detailed analysis was done before

In a number of previous postings I have presented various statistical measures describing returns and risk of my portfolio. I have computed some very detailed statistics such as the average daily return and the sample standard deviation of daily returns. I also looked at the correlation of returns on my portfolio with those of the S&P 500 index.

The reason I’ve done all this is because at that time I had no other objective measure to judge performance. If one could produce an annual return of 30% year after year, needless to say, this would be an outstanding strategy and an easy sale to any investor. However, this is hard to achieve. So, for example, I made a comparison to the S&P 500 index.

My trading strategy most closely matches that of a market-neutral hedge fund. This is because, as long as the strategy is being followed, there is in some sense (under a certain model) no exposure to small market moves, whether upward or downward. Therefore, it is interesting to compare its performance to that of hedge funds that declare that they pursue a strategy of the same general type.

I have done a search for publicly available hedge fund performance measures and found the following relevant pages:

• EDHEC Alternative Indexes produced by EDHEC-RISK, an Asset Management Research arm of EDHEC business school, rated one of top five in France.
• Dow Jones Hedge Fund Indexes, produced by Dow Jones & Company, Inc..
• HFRI Monthly Performance Indices, produced by Hedge Fund Research, Inc.

Understanding EDHEC Alternative Indexes

EDHEC Alternative Indexes gives values for “Annual Average Return since January 2001″ and also “Annual Std Dev since January 2001″. These values are given for a variety of types of trading strategies. For “equity market neutral” strategy type, the annual average return is given as 6.22% and the annualized standard deviation is 1.42%. The latter appears to be computed as the sample standard deviation of the series of monthly returns, annualized by multiplying it by the factor of the square root of 12 (the number of months in a year).

EDHEC also makes available one-page PDF summaries, one per strategy type, which are useful in understanding the origins of their numbers. Compared to the page, comparing performance of indices for different strategies, these summaries use a somewhat longer time frame for comparison. They used data starting from sometime in 2000, so the average S&P 500 return from that time until now is very low (the index didn’t do well in 2001 and 2002). In contrast, the equity market-neutral index appears to do very well.

When I look at various sources for measures of hedge fund performance, I am looking for two measures - a measure of annual return and a measure of noise incurred in attaining the said returns. If I had both, I could compute similar measures for the series of returns on my portfolio and perform an apples-to-apples comparison. In case of EDHEC numbers, the quoted value of the annualized standard deviation of monthly returns, 1.42%, is very low in comparison to 13.93% value for the S&P 500 index. However, I was able to recompute the value of 1.42% from the series of monthly returns EDHEC makes available for download. I believe that the reason for such a low measure of noise is twofold. First, any index represents some generalized average of its components. Hence the index will be less volatile than any given hedge fund that is one of its components. The other reason might stem from the particular method used by EDHEC. I have seen a mention in one of the research articles on the EDHEC site of the averaging methods used by different sources (such as Dow Jones, HFRI, the EDHEC itself and others). For example, two popular methods are capital-weighted averaging and unweighted averaging. In that article, EDHEC was said to use the method of principal components. I think this causes the EDHEC index to be an especially low-noise measure of the average return for a particular strategy. However, it also causes it to be useless for understanding the noise (annualized standard deviation) of returns of any particular hedge fund.

Understanding Dow Jones Hedge Fund Indices

The site of Dow Jones Hedge Fund Indexes currently shows partial (ending November 2006) performance figures for 2006 as well as the full-year performance figures for years 2002-2005. The index returns are given for six strategy types:

• Convertible arbitrage
• Distressed Securities
• Equity Long/Short
• Equity Market Neutral
• Event Driven
• Merger Arbitrage

As before, I am primarily interested in comparing to the Equity Market Neutral Index. It is also interesting to see how other strategy types have performed.

DJ quotes the return for Equity Market Neutral Strategy for 12 months ending November 30, 2006 at 6.19%. This is quite close to the return of 6.34% given by EDHEC for the same strategy type for 2006 up to the end of November. It is nice to see some agreement between index providers.

According to Dow Jones Hedge Fund Indices, the best return during 12 months ending in November 2006 among all tracked strategy types was 14.57%. It was earned by the Distressed Securities index. It is nice to note that my returns (29% overall; 23% for the stock and option portfolio) have exceeded even this value.

Components of Dow Jones Hedge Fund Indices

The Down Jones site also has some press releases which shed light on companies used as components of Dow Jones Hedge Fund Indices. For example, a press release in August 2005 announced a removal of one company from the Equity Market Neutral Strategy Benchmark:
Effective with the opening of trading on July 1, 2005, American Express Asset Management Group, Inc., will no longer be a component of the Benchmark. This press release also disclosed the typical number of hedge funds used as components of a single strategy index: It will not be replaced immediately, leaving the Equity Market Neutral strategy with six managers as components.

Another announcement disclosing changes in index components came out in January 2006. Some interesting names were mentioned. Focusing on the Equity Market Neutral strategy, two companies - Numeric Investors, L.P. and State Street Global Advisors - were removed from the index and one company, Analytic Investors, Inc., was mentioned as being added to the index. The same release mentioned that a total of 34 funds were being used as components of six hedge fund indices. The point is that the number of funds that determine the indices is not large at all.

A press release in April 2006 gave another example of an index component. A company called SSI Investment Management Inc. was being removed from Equity Market Neutral benchmark.

Summary

It appears that my portfolio has performed quite well compared to hedge fund indices tracked by Dow Jones Hedge Fund Indices, Inc. and EDHEC-RISK. I plan to continue the comparison in a bit more detail in a future posting.

It seems that I am currently in agreement with the market, as the differences of model and market prices are not great enough to result in new trade ideas. I hope this will change as the January earnings season approaches along with greater price moves and, hopefully, with prices getting out of alignment.

How to you know that you are earning good returns?

Earning 28% per year does sound good, compared to about 15.7% for the S&P 500 index and perhaps under 5% for a savings account. But then there is also the issue of noise in the growth of the portfolio. The value of a savings account just keeps growing and growing; there is no noise at all. Aside from a small chance of a bank’s bankruptcy and an FDIC intervention, it does not normally undergo any dramatic events. What about my portfolios and the S&P 500 index?

Investing in a single stock, or a broad-based index such as S&P 500, or a certain trading strategy or a fund manager brings about volatility in the path taken by the value of the account from the beginning of the year to its end. For instance, the highest daily return of S&P 500 in 2006 was 2.14% and the lowest return was -1.84%. So, a relevant question might be: what daily drop in the value of the portfolio would you be able to tolerate without responding to pain and going into less risky assets, such as bonds or even a savings account (which is essentially equivalent to investing in short-term money market instruments)?

The highest and lowest daily returns, for my entire portfolio in 2006, were approximately +4% and -4%, respectively. For the stock and option portfolio, they were roughly +5% and -5%, respectively. Both portfolios were “more noisy” than the S&P 500 index.

Designing an annual performance measure

In previous postings I have discussed a daily performance measure, an adaptation of the Sharpe Ratio concept, defined as the ratio of the average daily return to a standard deviation of the daily return. This measure does reward higher returns, as well as punishing noisier trading strategies, so it does reflect the risk-aversion of an educated speculator. However, most people are unaccustomed to thinking of average daily returns - people commonly think of annual returns instead. Looking at the sample standard deviation of a daily return is comparatively more useful. One would have to have a long - decades long - series of data to compute the standard deviation of annual returns. Such data do not exist for any fund manager. Using daily returns, and computing a sample standard deviation is hence the best substitute.

To create an easier understood performance measure, I will be looking to use an annual return.. As for a measure of noise, I will use the standard deviation of a daily return, scaled by the square root of the number of trading days in a year, assumed to be 250. The “Annualized Sharpe Ratio” is hence defined as (Annual Return - Risk-Free Rate) / (Standard Deviation of Daily Return * sqrt(Business Days Per Year)).

The above definition uses the textbook concept of a risk-free interest rate. It was not needed when the Sharpe Ratio was being computed on a daily scale. However, it does matter on an annual scale. If one wanted to leverage a position in the index, by borrowing cash and investing it in more than a 100%-weighted position in the index, one would be paying the borrowing cost and this is how an interest rate comes in. A realistic borrowing cost would likely be higher than the figures I assume, but this depends on the borrower. I assume this rate to be 5% for year 2006 and 3% for the entire trading history. These figures are chosen arbitrarily, but are probably in the right ballpark for typical short-term government yields, averaged over 2006 and over the last 4 years.

The conclusion from the above table is that the stock-and-option portfolio has outperformed the S&P 500 index in 2006, as judged by the Annualized Sharpe Ratio being 1.43 vs. 1.06. Less significantly (as futures trading is not meaningful and should not be considered in a fair comparison), the full portfolio has also outperformed the index: 1.16 vs. 1.06.

It would be very interesting to compare the Sharpe Ratios for many different portfolio managers and rank them accordingly. This would be a suitable way to rank my performance. However, the requisite information - namely, any sort of measure of the noise sustained by the portfolio as it attains a publicly disclosed annual return - is typically unavailable. So this comparison, while potentially very informative, cannot be done.

Annualized Sharpe Ratio for the entire trading history

The performance figures for my portfolio have improved over time. So, the figures given above for 2006 are better than figures computed for the entire trading history. To compute the latter, I define the
“Annualized Sharpe Ratio” as ((Business Days Per Year) * (Average Daily Return) - Risk-Free Rate) / (Standard Deviation of Daily Return * sqrt(Business Days Per Year)).

Compared to the definition of the Annualized Sharpe Ratio for 2006, this definition uses the number of business days in a year (assumed to be 250), multiplied by the average daily return, whereas the definition for year 2006 simply used the actual annual return figure for 2006. The results are:

Not very surprisingly, the results show that, considering the entire trading history, the full portfolio and also the stock-and-option part of the portfolio have both underperformed the S&P 500 index. This is just the other side of the coin. Although the performance improved in 2006, indeed, resulting in me outperforming the index, the prior years 2005 and 2004 were significantly noisier.

2006 was a fairly good year. The whole portfolio earned about 29%, while the stock-and-option trading earned 23%. In comparison, a holder of the S&P 500 index would have earned 15.7% (this figure includes the dividend yield, assumed to be 1.79% p.a. as estimated by Prof. Aswath Damodaran of NYU).

I have previously presented the summary statistics for the series of (logarithmic, or continuously compounded) daily returns on the whole portfolio, and the portfolio ex-futures. An update to those figures is now in order.

The “Average” row shows the average daily return. The row labeled “St.Dev.” gives the sample standard deviation of the series of about 250 (the number of trading days in one year) daily returns. The last row, “Sharpe Ratio” gives the ratio of average to standard deviation. These statistics are computed for three investment strategies. The “Full” column shows statistics for the entire portfolio, including trades in stocks, stock options and futures. As trading in futures is not particularly meaningful, and is done for the peace of mind only, its contribution to P&L of the whole portfolio can be isolated, leaving just the stock and option trades. Statistical measures for the resulting series of returns are shown in column “Ex-Futures”. The last column shows statistics for the S&P 500 index (each daily return has been incremented by a daily share of the assumed 1.79% per year dividend yield).

The conclusion is that the higher return, 0.083% daily, of my stock and option trading, as compared to 0.058% for the index, has come at the expense of a higher daily noise of 0.79% as compared to 0.63% for the index. However, when the ratio of the average daily return to its standard deviation is taken as the Sharpe Ratio measure of performance, it is seen that I have managed to outperform the index (the value of this ratio being 0.104 for the stock-and-option portfolio versus the value of 0.092 for the index).

The value of one dollar, invested as of January 1, 2006, in (dark blue) my actual portfolio, (purple) my portfolio with the exception of all futures trades and (yellow) the S&P 500 index:

Imagine that one dollar was invested on January 1, 2006. The value of the resulting portfolio during the year is shown by the dark blue line in the above chart. This line is built without any assumptions; it is merely a rescaled chart of the actual dollar value of the account. Positions were marked at mid-market to arrive at this value.

As the trading of futures is not done according to any model, it makes sense to isolate its effect and consider the portfolio value net of the contribution of trades in futures. The value of the one dollar under this assumption is shown by the purple line. The series, plotted by this line, is a series of mark-to-market values of a portfolio that includes trades in stocks, stock options, interest amounts, stock splits and dividend payments. The mark-to-market calculation is done using the daily closing prices I maintain in a database.

Lastly, the yellow line represents the value of one dollar invested in the S&P 500 index. As this index is not adjusted for dividends, I make an assumption about the annual dividend yield. Prof. Aswath Damodaran of NYU gives an estimate of 1.79% annual yield. Accordingly, each daily return of the series of index returns, used to build the yellow line, is incremented by 1.79%/250, assuming 250 business days per year.

The following table shows the same statistics for the entire trading history, up to the end of the year 2006.

The following chart is similar to the chart for 2006 above, except it covers the entire trading history. It shows the value of one dollar, invested as of the date the trading commenced, in (dark blue) my actual portfolio, (purple) my portfolio with the exception of all futures trades and (yellow) the S&P 500 index.

COF Update

The morning of the Boxing Day (the day after Christmas) I saw the widest spreads in COF that I’ve even seen: 10 cents or maybe even 15 cents bid-ask spreads in the first 15 minutes following NYSE open. Curiously, the option prices haven’t widened proportionally. I normally expect to see usually 5 cent and sometimes 10 cent spreads for options priced around a dollar. Today’s morning they were wider, but not by as much: I saw spreads of 10 cents and occasionally 15 cents.

Clearly, trading at such wide spreads does not make sense. While somewhat tighter spreads were seen during the trading days of the rest of the week, little trading was done indeed and the stock remained range bound.

January 3 was the first trading day of the New Year. Wide spreads of about 5 cents were seen in the morning for quite a while after the open, but the spread tightened later on.

COF Update

The most notable development was a notable decline in liquidity. The typical bid-ask spread observed during the day has widened from 1 cent to about 2 cents. However, the jumpiness of price moves has risen by more than this. The sharp price moves just after the open at 9:30 AM show the decline in liquidity most spectacularly.

For example, a stop order set at, e.g., 77.50 and trigered shortly after the open, could be executed as far as 20 cents away from the trigger level. The points to note are that,

• Stop orders are not normally executed outside of NYSE hours, as the stock price is not really well-defined (the bid-ask spread is very large). It is only in case of exceptional circumstances (an earnings release, for example), that the underlying risk that led to stop orders is sometimes hedged manually.
• The observation that the order was triggered after the open means that just before the open or around the opening time, the prices have not yet moved enough to trigger the stop. In other words, it is not the case that the execution of the order is delayed by the decision to not hedge prior to the open.

So, it is really the widened bid-ask spread or a very jumpy price behavior,
that lead to this exceptionally large slippage.

Throughout the past week, COF continued to trade in a range, and no exceptional price moves have taken place. The model-based estimate of the current realized volatility has remained at about 25-25.5%.

The past week has been relatively quiet. COF has been range-bound, trading between 75.75 and 78.74. At Friday’s close it was about 1% down from its level when the current strategy was initiated. Its intraday volatility was largely as expected. I would have benefited from setting wider intervals for delta hedging (at the expense of a larger tracking error if any big move did take place).

No new ideas have come up, as the market scanner’s analysis of volatility measures continues rejecting all supported strategies. I would like to spend some time working on new model development; this is the best way to generate ideas.

In other markets, I have noticed a significant decline of EUR and JPY against the dollar on Friday.

For several weeks, my market scanning program was returning exactly zero new trade ideas. In other words, all strategies of several types the program supports were being rejected. I am pleased to report that a new strategy has appeared and has been put in action. The idea: shorting the volatility of COF (Capital One Financial Corp.), using the options expiring in March 2007. Let’s talk about some basic features of this stock and its options.

COF is a fairly expensive stock that traded around 77.50 recently. It is fairly liquid. While I was watching the market and putting positions in place, the bid-ask spread in the stock price was 1 cent most of the time, only occasionally widening to 2 cents. However, the prices often moved with a sort of a jump, i.e., from 77.45-46 bid-ask, to 77.47-48 (as opposed to moving to, e.g., 77.46-47), so the true average sum of slippage and spread is probably a bit over 1 cent. This jumpiness is bad when trading the hedge position.

The March 2007 options on COF displayed varying liquidity over time. I am typically interested in options that are somewhat out-of-the-money. For options priced at around 1.50-2.00, the bid-ask varied from very occasional 5 cents to all the way wide at 15 cents, while I was monitoring them. This corresponds to a widening of bid-ask spread in implied vol from about 0.3% to about 1%. The spread of 10 cents, or about 0.6% in terms of implied vol, was most typical. So, it was important to trade carefully.

The main option being shorted had the implied volatility of about 28.5% when the pre-trade analysis was done. I saw the current (realized) volatility at about 23%. An analysis of recent history suggests that, in the longer term in the past, the typical volatility was about 22%.

COF last released its earnings on October 18, 2006. The next quarterly release will probably come around January 18, 2007 and the following one, around April 18, 2007, will occur after the March 2007 options expire. So there is one earnings release between now and the expiration date.

Needless to say (this is beyond the scope of this post), the options traded also pass other criteria - particularly the criteria, comparing the model-based prices to prices seen in the market.

Disclaimer: I hold positions in securities discussed in this posting. This posting is not an investment advice.

Today I will analyze the correlation of contemporaneous returns on my portfolio with those of S&P 500 index. Just as I would have hoped, the correlation between returns of the stock-and-option portfolio and those of the index has been close to zero since when I started using my current set of models.

The input data are prepared as follows.

A series of dates is considered. This series covers all business days since late 2003 and until now. Prior to this, these were some missing data. The returns for the whole portfolio, the portfolio excluding the impact of trades in futures, and the S&P 500 index are computed for consecutive pairs of dates. I then compute rolling correlations. For a given date, I take a window of the series of returns that includes the preceding 6 months, take the series of index returns and returns for either the full portfolio, or the portfolio ex-futures, and compute the sample correlation.

The result is an estimate of correlation between daily returns, on the same day, of the index and of my portfolio (or the portfolio ex-futures). The estimates are available from early 2004 (the dates of the first 6 months are lost because to estimate the correlation for a given date, I need to have data for all of the preceding 6 months) until present. The results are shown in the following chart.

Reviewing the above chart, I cannot explain the strongly negative correlations seen at the end of 2004 and the start of 2005. To understand this, I would need to review the positions at that time - for example, whether they were balanced between puts and calls or whether one option kind was traded in preference to the other. I think this negative correlation was due to an outright short position (not short volatility - this would not be unusual at all, but actually short one or more stocks). This position or rather, some trading that preceded me taking this position, were in violation of the rules that I impose on the type of positions that I allow myself to take.

Since then, I have been much more diligent in avoiding directional positions in stocks. As the chart shows, the correlation (for the stock-and-option portfolio) was quite small since the middle of 2005. It has been positive, which is not very good. However, this reflects the basic fact that a portfolio short in volatility tends to decline in value, as the volatility rises when the market experiences a decline. This positive correlation is hard to avoid.

To explain the positive and not-so-small (around 20% during the recent months) correlation between the returns of the whole portfolio and the index, I would need to research the correlation between the major foreign currencies and S&P 500. This is not a very interesting subject in the context of my typical positions. For example, if the Euro happened to be somewhat correlated with S&P 500, this would not mean much as I do not trade any pure correlation products. If I were aggressive in utilizing margin or the Value-at-Risk allotment for the entire portfolio, then the co-dependency between futures and stocks would be of importance. However, the risks being taken are quite moderate (I wish there were more good trade ideas!) In the long run this correlation simply does not matter.

Summary

I have observed that the correlation between my trading in stocks and options has been small and positive over the latest year. Its small magnitude is reasonable, as most of the time the positions are delta-neutral. The small positive value might be due to mis-hedging - meaning that I need to hold small short-the-stocks positions in addition to whatever hedges my models tell me to hold. Alternatively, it might be the evidence of me not buying insurance against a sudden drop in the market. Time will tell.